Journal of Economic Dynamics & Control

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1 Journal of Economic Dynamics & Control 36 (2012) Contents lists available at SciVerse ScienceDirect Journal of Economic Dynamics & Control journal homepage: The dynamics of mergers and acquisitions in oligopolistic industries Dirk Hackbarth a,n, Jianjun Miao b,c a Department of Finance, University of Illinois, 515 East Gregory Drive, Champaign, IL 61820, USA b Department of Economics, Boston University, 270 Bay State Road, Boston, MA 02215, USA c CEMA, Central University of Finance and Economics, and AFR, Zhejiang University, China article info Article history: Received 27 February 2011 Received in revised form 20 September 2011 Accepted 11 November 2011 Available online 9 December 2011 JEL classification: G13 G31 G34 abstract This paper embeds an oligopolistic industry structure in a real options framework in which synergy gains of horizontal mergers arise endogenously and vary stochastically over time. We find that (i) mergers are more likely in more concentrated industries; (ii) mergers are more likely in industries that are more exposed to industry-wide shocks; (iii) returns to merger and rival firms arising from restructuring are higher in more concentrated industries; (iv) increased industry competition delays the timing of mergers; (v) in sufficiently concentrated industries, bidder competition induces a bid premium that declines with product market competition; and (vi) mergers are more likely and yield larger returns in industries with higher dispersion in firm size. & 2011 Elsevier B.V. All rights reserved. Keywords: Anticompetitive effect Industry structure Real options Takeovers 1. Introduction Despite extensive research on mergers and acquisitions, some important issues in the takeover process remain unclear. Most models tend to focus on firm characteristics to explain why firms should merge or restructure and do not endogenize the timing and terms of a takeover deal or the synergy gains arising from a deal. In addition, most real options models of investment under uncertainty consider symmetric firms in a competitive industry when they study product market competition (see, e.g., Dixit and Pindyck, 1994; Miao, 2005), while real options models of mergers and acquisitions typically abstract from industry characteristics. On the other hand, most studies in the industrial organization literature (see, e.g., Salant et al., 1983; Perry and Porter, 1985; Farrell and Shapiro, 1990) build models of exogenous mergers to examine their welfare implications under different policy settings. Yet little work has been done on the implications of industrial organization for the dynamics of mergers. Thus, an important and heretofore overlooked aspect of bid premiums, merger returns, terms, and timing is their relation to industrial organization. We embed takeovers in an industry equilibrium model with heterogeneous firms that allows us to study the effect of strategic product market considerations on: (i) the endogenous synergy gains from merging; (ii) the joint determination of the timing and terms of mergers; and (iii) the returns to both merging and rival firms. 1 The model also provides novel insights into how the interaction between industry competition and bidder competition for a scarce target influences bid premiums in takeover deals. To our knowledge, this paper presents the first unified framework to examine the link between n Corresponding author. Tel.: þ ; fax: þ addresses: dhackbar@illinois.edu (D. Hackbarth), miaoj@bu.edu (J. Miao). 1 Weston et al. (2003) provide a detailed review of the literature on mergers and acquisitions /$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi: /j.jedc

2 586 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) firm heterogeneity, industry structure, and the timing, terms, and returns of takeovers, and to relate the outcome of takeover contests with multiple bidders to product market competition. There is ample empirical evidence on industry characteristics affecting mergers and acquisitions in practice. Mitchell and Mulherin (1996), Andrade et al. (2001), Andrade and Stafford (2004), and Harford (2005) find that takeover activity is driven by industry-wide shocks, and is strongly clustered by industry. Borenstein (1990) and Kim and Singal (1993) document that airline fares on routes affected by a merger increase significantly over those on routes not affected by a merger. Kim and Singal (1993) and Singal (1996) also identify a positive relation between airfares and industry concentration. Eckbo (1985) documents that acquirers and targets earn positive returns, while Singal (1996) reports in addition that returns to rival firms are positively related to changes in industry concentration. This non-exhaustive list of empirical research strongly supports the notion that mergers and acquisitions not only affect product market outcomes and vice versa but also have substantial anticompetitive effects. 2 Against the backdrop of the industry-level evidence, we study the timing of a horizontal merger of two firms in an asymmetric industry equilibrium. To analyze the role of industry characteristics for the takeover process, we embed an oligopolistic industry structure similar to Perry and Porter s (1985) in a dynamic real options model of mergers and acquisitions. As a result, synergy gains arise endogenously due to oligopolistic (Cournot) competition. 3 In particular, we specify a tangible asset that helps firms to produce output at a given average cost. The merged entity operates the tangible assets of the two merging partners, so it is larger and its average and marginal costs are lower. This allows us to capture economically relevant and realistic asymmetries caused by the merger of a subset of firms. In addition to a different cost structure for the merged firm, all firms face a different industry structure after a merger of two firms. Recognizing these economic effects, we derive closed-form solutions to industry equilibria in a real options model of takeovers in which the merger benefits, the merger timing, and the merger terms are determined endogenously based on an option exercise game featuring both cost reduction and changes in product market competition. Our model delivers a number of predictions that are in line with the available evidence. First, we analyze shocks to industry demand, which relax the assumption of an exogenously given market size in previous analyses. For our model with a stochastic market size, we demonstrate that a merger occurs the first time the demand shock hits a trigger value from below. In other words, control transactions emerge as an endogenous product market outcome from rising demand shocks in the model and hence cyclical product markets generate procyclical takeover activity. Second, our real options model formalizes the folklore that mergers are more likely in industries that are more exposed to or more sensitive to industry-wide demand shocks. All of these implications are consistent with the evidence reported by, e.g., Mitchell and Mulherin (1996) and Maksimovic and Phillips (2001). Third, cumulative stock returns are higher for the smaller merging firm (e.g. target) than the larger merging firm (e.g. acquirer) when the firms have identical merger costs. The intuition for this interesting finding is that the smaller firm benefits relatively more from a merger in our industry equilibrium and hence enjoys a larger relative synergy gain (i.e. return). Because mergers typically involve acquisitions of a small firm by a large firm (see, e.g., Andrade et al., 2001 or Moeller et al., 2004), this implication supports the available evidence without reliance on additional assumptions, such as asymmetric information or misvaluation. Moreover, our model has several novel testable implications regarding the timing and terms of mergers, the returns to merging firms, and the bid premium in contested deals, which demonstrate the importance of relaxing the assumption of exogenous synergy gains, such as scale economies. First, the model s industry equilibrium reveals that cumulative stock returns are determined by an anticompetitive effect, two size effects, and a hysteresis effect. While the latter two effects are shared by other real options models of mergers, the anticompetitive effect is unique to our analysis. 4 Due to this effect, we find that returns to merging and rival firms arising from restructuring are higher in more concentrated industries. Intuitively, returns are under certain conditions positively related to anticompetitive profit gains, which are positively related to industry concentration. Second, and perhaps surprisingly, increased product market competition delays the timing of mergers in our real options model. Notably, this result is contrary to some conclusions in recent research on irreversible investment under uncertainty. Grenadier (2002), for instance, emphasizes that more industry competition increases the opportunity cost of waiting to invest and thus accelerates option exercise. In his symmetric industry model, anticompetitive profits result from exogenously reducing the number of identical firms that compete in the industry. Firms are allowed to be asymmetric in our model, however, and anticompetitive profits result from endogenous synergy gains of combining two firms. Crucially, these synergy 2 Eckbo (1983), Eckbo (1985), and Eckbo and Wier (1985) find little evidence that challenged horizontal mergers are anticompetitive. McAfee and Williams (1988) question whether event studies can detect anticompetitive mergers. Andrade and Stafford (2004), Holmstrom and Kaplan (2001), Shleifer and Vishny (2003) differentiate the takeover activity in the 1960s and 1970s and the more recent merger waves of the 1980s and 1990s. 3 Unlike earlier real options models of mergers, we do not assume exogenous synergy gains from merging, such as economies of scale or efficiencyenhancing capital reallocation. There are several recent cases of mergers that match our modeling approach closely. These cases were investigated by the Department of Justice and approved if they were not likely to reduce competition substantially despite of the substantial change of the cost structure and the industry structure. For example, Whirlpool acquired Maytag in 2006 and gained market power in the appliance industry. Whirlpool also substantiated large cost savings. Another example is the $2.6 billion deal of 2008 between two major US air carriers, Delta and Northwest. This merger reduced the combined carrier s operating costs without being viewed to damage competition substantially. 4 The anticompetitive effect also affects the merger terms. In Section 4.1, we show that a large acquirer (small target) demands a lower (higher) ownership share in the merged firm if the industry is less competitive.

3 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) gains are lower in more competitive industries, ceteris paribus, because a merger of two firms has a smaller effect on output price increases in these industries. These lower synergy gains from exercising the merger option offset the opportunity cost of waiting to merge. Thus, by relaxing some key features of previous work, our real options model implies that firms in more competitive industries may optimally exercise their option to merge later. Third, our model sheds light on if and how multiple sources of competition (i.e. bidder competition and industry competition) in our real options framework influence bid premiums in control transactions. Intuitively, bidder competition puts the target in an advantageous position and allows target shareholders to extract a higher premium from the bidding firms. Hence one might be tempted to conclude that bidder competition generally raises the price of a contested deal irrespective of industry competition. However, we demonstrate that in our model the degree of industry competition plays a central role for if and how a bid premium emerges in equilibrium. Suppose a large firm bidder and a small firm bidder compete for a small firm target. When their merger costs are similar, the large firm bidder wins the takeover contest, which is consistent with the stylized fact that targets are on average smaller than acquirers. Notably, only in a sufficiently concentrated industry, bidder competition speeds up the takeover process and leads the large firm acquirer to pay a bid premium to discourage the competing small firm competitor. This bid premium decreases with industry competition. By contrast, in a more competitive industry, the small firm bidder does not matter, and the equilibrium of the option exercise game corresponds to the one without bidder competition. If the two bidding firms are identical, then bidder competition will squeeze away all surplus to the bidders and allow target shareholders to extract all merger surplus. Fourth, our analysis provides new insights into the role of dispersion in firm size for timing and returns of mergers. We find that, all else equal, takeovers are more likely in industries in which firm size is more dispersed. In our industry framework, the endogenous synergy gains from merging increase when the size difference of the two merging partners tangible assets rises. Because of this economic effect, the model also predicts that returns to merging and rival firms arising from restructuring are higher in industries with more dispersion in firm size. Our work contributes to a growing body of research using the real options approach to analyze mergers and acquisitions. Lambrecht (2004), Morellec and Zhdanov (2005), Hackbarth and Morellec (2008), Margsiri et al. (2008), and Bernile et al. (in press) analyze the timing and terms of takeovers under exogenous synergy gains in that takeovers provide either economies of scale or result in a more efficient allocation of resources. To isolate the effect of industry competition on mergers as much as possible, our model abstracts from these exogenous synergy gains. Lambrecht briefly analyzes a duopoly model and shows that market power enhances symmetric firms exogenous synergy gains. In contrast, we consider an oligopolistic Cournot Nash equilibrium with an arbitrary number of asymmetric firms competing with each other in the product market which allows for endogenous synergy gains. More recently, Leland (2007) considers purely financial synergies in motivating acquisitions when timing is exogenous, while Morellec and Zhdanov (2008) explore interactions between financial leverage and takeover activity with endogenous timing. None of these studies considers the relation of oligopolistic industry equilibria and takeover activity. Lastly, our paper is also related to Grenadier (2002) who studies the effects of industry competition on the exercise of real options in oligopolistic industries. But he does not study takeovers. Other complementary but less related models study industry dynamics and linkages between incentives to merge and entry or investment; see especially Gowrisankaran and Holmes (2004) or Pesendorfer (2005) and Qiu and Zhou (2007). Unlike our study, these authors typically focus on welfare implications of mergers as well as long-run industry equilibria (i.e., if mergers towards monopoly are profitable or not). Instead, we focus on the merger timing equilibrium of two firms in an industry without entry where the merger in turn affects industry equilibrium. Our paper is also related to Gorton et al. (2009), who propose a theory of mergers that combines managerial merger motives with an industry level regime shift that may lead to some value-increasing merger opportunities. They also emphasize the importance of the distribution of firm size in an industry. Unlike our analysis, they do not consider industry competition and its interaction with bidder competition in a dynamic environment. All these articles are silent on the role of industry structure for merger timing, terms, and returns, or bid premiums, which are central to our analysis. The remainder of the article is organized as follows. Section 2 presents the model. Section 3 derives industry equilibrium without merger and analyzes the incentive to merge. Section 4 studies equilibrium mergers with a single bidder, while Section 5 studies equilibrium mergers with multiple bidders. Section 6 concludes. Proofs are relegated to Appendix A. 2. The model We consider an industry with infinitely-lived firms whose assets generate a continuous stream of cash flows. The industry consists of N heterogeneous firms that produce a single homogeneous product, where N Z2 is an integer. Each firm i initially owns an amount k i 40 of physical capital. To focus on the timing and returns of mergers and acquisitions and to keep the industry analysis tractable, we follow Perry and Porter (1985), Shleifer and Vishny (2003), and others by assuming that firms can grow through takeovers, but not through internal investments. 5 In addition, we assume that productive capital does not depreciate over time. The industry s total capital stock is in fixed supply and equal to K. Therefore, the industry s capital stock at each 5 See e.g. Bernile et al. (in press) on takeovers and entry or Margsiri et al. (2008) on takeovers and internal growth.

4 588 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) time t satisfies X N i ¼ 1 k i ¼ K: The cost structure is important in the model. We denote by Cðq,k i Þ the cost function of a firm that owns an amount k i of the capital stock and produces output q. The output q is produced with a combination of the fixed capital input, k i, and a vector of variable inputs, z, according to a smooth concave production function, q ¼ Fðz,k i Þ. Then the cost function Cðq,k i Þ is obtained from the cost minimization problem. To isolate the role of product market competition on mergers, assume that the production function F has constant returns to scale. This implies that Cðq,k i Þ is linearly homogeneous in ðq,k i Þ. For analytical tractability, we adopt the quadratic specification p of the cost function, Cðq,k i Þ¼ 1 2 q2 =k i. This cost function may result from the Cobb Douglas production function q ¼ ffiffiffiffiffiffi k i z, where z may represent labor input. Note that both the average and marginal costs decline with the capital asset k i. Salant et al. (1983) show that if average cost is constant and independent of firm size, a merger may be unprofitable in a Cournot oligopoly with linear demand. It is profitable if and only if duopolists merge into monopoly. As Perry and Porter (1985) point out, the constant average cost assumption does not provide a sensible description of mergers. The productive capital (or tangible asset) plays an important role in the model. It allows us to capture economically relevant and realistic asymmetries caused by the merger of a subset of heterogeneous firms. Notably, the merged entity faces a different optimization problem as a result of a change in production costs (i.e., average and marginal costs decline with the amount of productive capital employed) and important strategic considerations (i.e., increased market power). We now integrate these effects into a dynamic real options model of mergers. To do so, let the industry s inverse demand at time t is given by the linear function PðtÞ¼aYðtÞ bqðtþ, ð2þ where Y(t) denotes the industry s demand shock at time t observed by all firms, Q(t) is the industry s output at time t, and a and b are positive constants; a represents exposure of demand to industry-wide shocks, and b represents the price sensitivity of demand. For all t 40, we assume that the industry s demand shock is governed by the geometric Brownian motion process dyðtþ¼myðtþ dt þsyðtþ dwðtþ, Yð0Þ¼y 0, ð3þ where m and s are constants and ðw t Þ t Z 0 is a standard Brownian motion defined on ðo,f,pþ. To ensure that the present value of profits is finite, we make the following assumption: Assumption 1. The parameters m, s, and r satisfy the condition 2ðmþs 2 =2Þor. Following Shleifer and Vishny (2003), we consider a subset of two firms i (i¼1,2) with capital stocks k i, which can negotiate a takeover deal at time t 40 if it is in their shareholders best interest. 6 To this end, we assume that each firm i s strategy space is restricted to the optimal exercise strategy of its merger option. For each firm i, this strategy is given by a threshold y n i,such that the merger is executed the first time when Y(t) exceeds y n i. The second-stage negotiation problem then reduces to identifying the merger terms, x i, which will induce both firms to exercise their merger option at the globally efficient merger threshold, y n. In reality, control transactions are costly. We therefore assume that each merging firm i incurs a fixed lump-sum cost X i 40fori¼1,2. This cost captures fees to investment banks and lawyers as well as the cost of restructuring. Finally, we assume that all firms in the industry are Cournot Nash players and that management acts in the best interests of shareholders. We also assume that shareholders are risk-neutral and discount cash flows by r 40. Thus, all decisions are rational and value-maximizing choices. 3. Industry equilibrium without merger To incorporate an oligopolistic industry structure into a real options framework of mergers, we need to work backwards. That is, we first characterize industry equilibrium when asymmetric firms play Cournot Nash strategies without mergers. This allows us to derive the firms value functions for a given industry structure. Next, we examine the incentive to merge, which is based on the merging firms net gains in these value functions resulting from a change in industry structure at an arbitrary time Equilibrium characterization and firm strategies ð1þ Let q i ðtþ denote the quantity selected by firm i at time t. Then firm i s instantaneous profit is given by p i ðtþ¼½ayðtþ bqðtþšq i ðtþ 1 2 q iðtþ 2 =k i, ð4þ 6 So far, few models endogenize multiple mergers of heterogenous firms (see, e.g., Qiu and Zhou, 2007). In a setup with symmetric firms, Kamien and Zang (1990) find that firms remain independent when merger decisions are endogenous. However, the multiplicity of equilibria in their model limits the robustness of their results. Similarly, our real options model can only produce reliable results for an exogenously assigned sequence of multiple mergers.

5 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) where QðtÞ¼ XN i ¼ 1 q i ðtþ is the industry output at time t. Given the instantaneous profits, we can compute firm value, or the present value of profits Z 1 V i ðyþ¼e y e rt p i ðtþ dt, ð6þ 0 where E y ½Š denotes the conditional expectation operator, given that the current industry shock takes the value Y(0)¼y. We define strategies and industry equilibrium as follows. The strategy fðq n 1 ðtþ,...,qn NðtÞÞ : t Z0g constitutes an industry (Markov perfect Nash) equilibrium if, given information available at date t, q n j ðtþ is optimal for firm j ¼ 1,...,N, when it takes other firms strategies q n i ðtþ for all iaj as given. Because firms play a game in continuous time, there could be multiple Markov perfect Nash equilibria, as is well known in game theory. Instead of finding all equilibria, we will focus on the equilibrium in which firms adopt static Cournot strategies. As is well known, the static Cournot strategies that firms play at each date constitute a Markov perfect Nash equilibrium. The first proposition characterizes this industry equilibrium. 7 Proposition 1. The strategy q n i ðtþ¼ y i ayðtþ 1þB b constitutes a Cournot Nash industry equilibrium at time t for firms i ¼ 1,...,N, where y i ¼ b bþk 1 i and B ¼ XN i ¼ 1 In this equilibrium, the industry output at time t is given by y i : Q n ðtþ¼ B ayðtþ 1þB b, ð9þ and the industry price at time t is given by P n ðtþ¼ ayðtþ 1þB : ð10þ ð5þ ð7þ ð8þ Note that this proposition also characterizes the equilibrium after a merger, once we change the number of firms and the capital stock of the merged firm. A merger brings the capital of two firms under a single authority and thus reduces production cost. We use the subscript M to denote the merged entity. The merged firm owns capital k M ¼ k 1 þk 2.ByProposition 1, the values of y i for the non-merging firms iz3 are unaffected by the merger. We can verify that the value of y for the merged firm satisfies maxfy 1,y 2 goy M ¼ y 1 þy 2 2y 1 y 2 oy 1 y 1 y 1 þy 2 : ð11þ 2 Thus, the value of B in Proposition 1, which determines total industry output, changes after the merger to B M ¼ Bþy M y 1 y 2 ob: ð12þ By Eqs. (9), (10), and (12), we conclude that the merger causes total output to fall and industry price to rise. In addition, we can use Eqs. (12) and (7) to show that maxfq n 1 ðtþ,qn 2 ðtþgoq MðtÞoq n 1 ðtþþqn 2 ðtþ, ð13þ where q M ðtþ is the output produced by the merged firm at time t. This result implies that the merged firm produces more than either of the two merging firms, but less than the total output level of the two. The analysis highlights the tension of a merger: after a merger, the industry price rises, but the merged firm restricts production. Thus, a merger may not always generate a profit gain. In order to analyze mergers tractably, we follow Perry and Porter (1985) and consider an oligopoly structure with small and large firms. Specifically, we assume that the industry initially consists of n identical large firms and m identical small firms. Each large firm owns an amount k of capital and incurs merger costs X l if it engages in a merger. Each small firm owns an amount k/2 of capital and incurs merger costs X s if it engages in a merger. In this case, Eq. (1) becomes nkþmk=2 ¼ K, and hence we can use m ¼ 2Kk 1 2n to replace m in the analysis below. We assume that either two small firms can merge (symmetric merger) or a small firm and a large firm can merge (asymmetric merger). In the former case, a merger preserves the two-type industry structure, but the number of small firms 7 Proofs for all propositions are given in Appendix A.

6 590 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) and large firms changes. In the latter case, firm heterogeneity increases. That is, all non-merging small or large firms remain identical, but the merged entity owns more physical capital than a large firm, destroying the two-type industry structure. We do not consider mergers between two large firms or further mergers over time. This assumption makes our analysis tractable and permits us to focus on the key questions of how product market competition interacts with bidder competition and how product market competition influences the timing and terms of mergers as well as merger returns. One justification of our assumption may be related to antitrust law. White (1987, p. 16) writes that: [the Horizontal Merger] Guidelines use the Herfindahl-Hirschman Index (HHI) as their primary market concentration guide, with concentration levels of 1,000 and 1,800 as their two key levels. Any merger in a market with a post-merger HHI below 1,000 is unlikely to be challenged; a merger in a market with a post-merger HHI above 1,800 is likely to be challenged (if the merger partners have market shares that cause the HHI to increase by more than 100), unless other mitigating circumstances exist, like easy entry. Mergers in markets with post-concentration HHI levels between 1,000 and 1,800 require further analysis before a decision is made whether to challenge. In our model, a merger of two large firms would raise industry concentration to a higher level than a merger of two small firms or a merger between a small firm and a large firm. The industry concentration level following a merger of two large firms is more likely to cross the regulatory threshold, and thus such a merger is more likely to be challenged by antitrust authorities. 8 As an example, the two largest office superstore chains in the United States, Office Depot and Staples, announced their agreement to merge on September 4, Seven months later, the Federal Trade Commission voted 4 to 1 to oppose the merger on the grounds that it was likely to harm competition and lead to higher prices in the market for the sale of consumable office supplies sold through office superstores. (see Dalkir and Warren-Boulton, 2004.) 3.2. Firm valuation and incentive to merge We first consider a symmetric merger of two small firms. Define DðnÞðbþk 1 Þðbþ2ðKbþ1Þk 1 Þ b 2 n40: Note that the argument n in this equation indicates that there are n large firms in the industry. This notation is useful for our merger analysis below because the number of large firms changes after a merger option is exercised. It follows from Eq. (1) that Kk 1 4n, which implies that both DðnÞ and Dðnþ1Þ are positive. We use Proposition 1 and Eq. (6) to compute firm values. Proposition 2. Consider a symmetric merger between two small firms in the small-large oligopoly industry. Suppose Assumption 1 holds. The equilibrium value of the type f ¼ s,l firm is given by where V f ðy; nþ¼ P f ðnþy 2 r 2ðmþs 2 =2Þ, P s ðnþ¼ a2 ðbþk 1 Þ 3 DðnÞ 2, ð16þ P l ðnþ¼ a2 ðbþ2k 1 Þ 2 ðbþk 1 =2Þ : ð17þ DðnÞ 2 After a merger between two small firms, there are nþ1 identical large firms and m 2 identical small firms in the industry. Thus, the value of the large firm after a merger is given by V l ðy; nþ1þ. It follows from Proposition 2 that the benefit from merging is given by V l ðy; nþ1þ 2V s ðy; nþ¼ ½P lðnþ1þ 2P s ðnþšy 2 : ð18þ r 2ðmþs 2 =2Þ The term P l ðnþ1þ 2P s ðnþ represents the profitability of an anticompetitive merger. For there to be takeover incentives, this term must be positive. After a merger, the number of small firms in the industry is reduced, and hence the industry s market structure is changed. As the output of two small firms prior to a merger exceeds the output of the merged firm, an incentive to merge requires that the increase in industry price be enough to offset the reduction in output of the merged entity. The conditions for takeover incentives may be summarized as follows. Proposition 3. Consider a symmetric merger between two small firms. Let DðnÞ be given in Eq. (14) and define the critical value ð14þ ð15þ D n b2 1 A, ð19þ 8 For an industry with two large firms and a price sensitivity of b¼0.5, our model s pre-merger HHI equals 1,378 when k¼0.2, K¼1, and a¼100. This concentration measure rises to 1,578 (1,734) following a merger of two small firms (a merger of a small and a large firm) compared to a post-merger HHI is 2,022 after a merger of two large firms. See Appendix A for derivations.

7 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) where A is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u A ðbþ2k 1 bþk 1 =2 Þt : ð20þ 2ðbþk 1 Þ 3 (i) If max DðnÞ¼Dð0ÞoD n, ð21þ n then there will always be an incentive to merge. (ii) If min DðnÞ¼DðK=k 1Þ4D n, ð22þ n then there will never be an incentive to merge. (iii) If Dð0Þ4D n 4DðK=k 1Þ, then when n is high enough so that DðnÞoD n, there will be an incentive to merge. In Proposition 3, (21) and (23) provide two conditions for takeover incentives in our Cournot Nash framework. 9 That is, when the increase in price outweighs the reduction in output so that the net effect leads to an increase in instantaneous profits, the two small firms have an incentive to form a large organization. These two conditions depend on the industry demand function through the price sensitivity b, and the size of a large firm k, and on the industry structure through the number n of large firms prior to the restructuring. Moreover, the proposition shows that there could be no incentives to merge if the condition in Eq. (22) holds. It is straightforward to show that there is always an incentive to merge when the industry consists of a small-firm duopoly. 10 A similar result is obtained by Perry and Porter (1985) and Salant et al. (1983) in a static industry model. Alternatively, after a merger between a large firm and a small firm, the industry consists of n 1 identical large firms, m 1 identical small firms, and a huge merged firm. The merged entity owns capital k M ¼ kþk=2 ¼ 3k=2. We use Proposition 1 and Eq. (6) to derive firm values. Proposition 4. Consider an asymmetric merger between a large firm and a small firm in the small-large oligopoly industry. Suppose Assumption 1 holds. After this merger, the equilibrium firm value is given by V a f ðy; n 1Þ¼ Pa f ðn 1Þy2 for f ¼ s,l,m, ð24þ r 2ðmþs 2 =2Þ where P a s ðn 1Þ¼ a 2 ðbþk 1 Þ 3 DðnÞ b 2 1þ 2 2, ð25þ P a l ðn 1Þ¼ a2 ðbþ2k 1 Þ 2 ðbþk 1 =2Þ DðnÞ b 2 1þ 2 2, ð26þ P a M ðn 1Þ¼ 3a2 ðbþ2k 1 Þ 2 ðbþk 1 Þ 2 ð3bþk 1 Þ=ð3bþ2k 1 Þ 2 DðnÞ b 2 1þ 2 2 : ð27þ ð23þ The benefit from an asymmetric merger is given by V a M ðy; n 1Þ V sðy; nþ V l ðy; nþ¼ ½Pa M ðn 1Þ P sðnþ P l ðnþšy 2 : ð28þ r 2ðmþs 2 =2Þ For there to be an incentive to merge, the expression in Eq. (28) must be positive. As in Proposition 3, we have the result. Proposition 5. Consider an asymmetric merger between a small firm and a large firm in the small-large oligopoly industry. Let DðnÞ be given in Eq. (14) and let the critical value D n take the value D a b2 1 D 1þ 2, ð29þ 9 Similar conditions have been derived by Perry and Porter (1985) in their static model. 10 One can verify that condition (21) is satisfied for k¼k, n¼0, and m¼2, which is a limiting case of our model.

8 592 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) where D is given by D ðbþk 1 Þðbþ2k 1 Þ 3bþ2k 1 Then parts (i) (iii) in Proposition 3 apply here. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u t 3ð3bþk 1 Þ : ð30þ ðbþk 1 Þ 3 þðbþ2k 1 Þ 2 ðbþk 1 =2Þ To facilitate the analysis for a single bidder, we make the following assumption. Assumption 2. Suppose P a M ðn 1Þ P sðnþ P l ðnþ40 and P l ðnþ1þ 2P s ðnþo0 so that there is an incentive to merge between a large firm and a small firm, but no incentive to merge between two small firms. A sufficient condition for this assumption is that case (i) or (iii) holds in Proposition 5, but both cases are violated in Proposition 3. We invoke an alternative assumption for the analysis of multiple bidders. Assumption 3. Suppose P l ðnþ1þ 2P s ðnþ40 and P a M ðn 1Þ P sðnþ P l ðnþ40 so that there is an incentive to merge between a large and a small firm and between two small firms. A sufficient condition for this assumption is that case (i) or (iii) holds in both Propositions 3 and 5. The following lemma is useful for our later merger analysis: Lemma 1. Under Assumptions 2 or 3, the profit differentials P a M ðn 1Þ P sðnþ P l ðnþ increase with the parameters a and n, and the profit ratios ½P a M ðn 1Þ P sðnþ P l ðnþš=p s ðnþ and ½P a M ðn 1Þ P sðnþ P l ðnþš=p l ðnþ increase with the parameter n. To interpret this lemma, recall that the parameter n represents the number of large firms in the industry prior to a merger. This parameter proxies for industry concentration. A higher value of n represents a higher level of industry concentration. The parameter a represents the exposure of industry demand to the industry-wide shock. A higher value of a implies that the increase in industry price is higher in response to an increase in the exogenous demand shock. Lemma 1 then demonstrates that under certain conditions the anticompetitive gains from a merger measured in terms of either profit differential or profit ratio are larger in more concentrated industries. In terms of profit differential, these gains are also greater in industries that are more exposed to industry-wide shocks. Example 1. The conditions in Assumption 2 or 3 are not easy to check analytically. We thus use a numerical example to illustrate them and Propositions 3 and 5. The baseline parameter values are: b¼0.5, k¼0.2, and K¼1. By Eq. (1), these values imply that the numbers of large and small firms must satisfy 2nþm¼10. Fig. 1 shows the profit differentials of symmetric and asymmetric mergers for a wide range of parameter values of b and n. Both profit differentials increase monotonically with the number of large firms n, but non-monotonically with the price sensitivity parameter b. Intuitively, this non-monotonicity results from two opposing effects of a decline in b; that is, it raises price, but reduces output. Thus, change in the price sensitivity parameter has an ambiguous effect on the profit differentials, because it depends on whether the price effect or the quantity effect dominates. From the lower left end of the surfaces in Fig. 1, we find that asymmetric mergers can be profitable, while symmetric mergers are not, so Assumption 2 is satisfied. This happens, for example, when b¼0.5, and n takesvaluesfrom1to5.when b¼0.4, and n takes values from 1 to 5, however, both symmetric and asymmetric mergers are profitable, so Assumption 3 is satisfied. Fig. 1. Profit differentials in symmetric and asymmetric mergers. This figure depicts the profit differentials P l ðnþ1þ 2P sðnþ and P a M ðn 1Þ PsðnÞ P lðnþ as a function of the price sensitivity parameter b and the number of large firms n when a¼100, K¼1, and k¼0.2.

9 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) Merger equilibrium with a single bidder After having characterized the industry equilibrium, we now derive the merger equilibrium for the case of a single bidder in this section and for the case of multiple bidders in the next section. In particular, we solve for the optimal merger timing and terms in both cases and compute merger returns and bid premiums Merger policies In this subsection, we analyze the timing and terms of a merger between a small firm target and a large firm bidder. We suppose Assumption 2 holds, so that two identical small firms do not have an incentive to merge, but a large firm and a small firm do have an incentive to merge. 11 We assume the acquirer submits a bid in the form of an ownership share of the merged firm s equity. Given this bid, both acquirer and target shareholders select their value-maximizing merger timing. In equilibrium, the merger timing chosen by the acquirer and the target are the same. The merger offers participants in the deal an option to exchange one asset for another. That is, they can exchange their shares in the initial firm for a fraction of the shares of the merged firm. Thus, the merger opportunity is analogous to an exchange option (Margrabe, 1978). The equilibrium timing and terms of the merger are the outcome of an option exercise game in which each participant determines an exercise strategy for its exchange option. We first solve for the equilibrium, and then show that the equilibrium timing is globally optimal. 12 To solve for the equilibrium, we first consider the exercise strategy of the large firm bidder. Let x l denote the ownership share of the large firm in the merged entity. Then 1 x l is the ownership share of the small firm target. The merger surplus accruing to the large firm is given by the positive part of the (net) payoff from the merger: ½x l V a M ðy; n 1Þ V lðy; nþ X l Š þ, where V a M ðy; n 1Þ and V lðy; nþ are given in Propositions 2 and 4. When it considers a merger, the large firm trades off the stochastic benefit from merging against the fixed cost X l of merging. Since firms have the option but not the obligation to merge, the surplus from merging has a call option feature. Let y n l denote the merger threshold selected by the large firm. The value of this firm s option to merge, denoted OM l ðy,y n,x l l; nþ for yry n l, is given by OM l ðy,y n l,x l; nþ¼e y fe rt y n l ½x l V a M ðyðt y n l Þ; n 1Þ V lðyðt y n l Þ; nþ X lšg, ð31þ where t y n denotes the first passage time of the process ðy l t Þ starting from the value y to the merger threshold y n l selected by the large firm. By a standard argument (e.g., Karatzas and Shreve, 1999), we can show that OM l ðy,y n l,x l; nþ¼½x l V a M ðyn l ; n 1Þ V lðy n l ; nþ X lš! b y, ð32þ y n l where b denotes the positive root of the characteristic equation 1 2 s2 bðb 1Þþmb r ¼ 0: ð33þ Note that it is straightforward to prove that b42 underassumption 1. Eq. (32) admits an intuitive interpretation. The value of the option to merge is equal to its share of the merger benefits net of the merger costs, ½x l V a M ðyn l ; n 1Þ V lðy n l ; nþ X lš, generated at the time of the merger multiplied by a discount factor ðy=y n l Þb. This discount factor can be interpreted as the Arrow Debreu price of a primary claim that delivers $1 at the time and in the state the merger occurs. The optimal threshold y n l selected by the large firm maximizes the value of the merger option in Eq. (32). Thus, it satisfies the first-order l ðy,y n l,x l; n l ¼ 0: Solving this equation yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y n l ðx l; nþ¼ bx l r 2ðmþs 2 =2Þ b 2 x l P a M ðn 1Þ P : lðnþ ð35þ ð34þ 11 This is reasonable because deals often involve a small firm being acquired by a large firm. In Andrade et al. (2001) sample of 4,256 deals over the period, the median target size is 11.7% of the size of the acquirer. Moeller et al. (2004) measure relative size as transaction value divided by acquirer s equity value, and report averages of 19.2% (50.2%) for 5,503 small (6,520 large) acquirers between 1980 and Our analysis follows similar steps as in Lambrecht s (2004) case of friendly mergers where both managements first decide on the timing of the merger and subsequently negotiate how to divide the merger gains.

10 594 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) Fig. 2. Reaction functions with a single bidder. The figure plots the reaction functions of the bidder and the target as a function of the bidder s ownership share. The decreasing (increasing) line represents the acquirer s (target s) strategy. The crossing point of the reaction functions represents the option exercise equilibrium of Proposition 6. It follows that, as a function of x l, y n l declines with x l. This function gives the merger threshold for a given value of the ownership share x l. Fig. 2 illustrates this function. We next turn to the exercise strategy of the small firm target, which can be solved in a similar fashion. The value of the small firm target s option to merge is given by OM s ðy,y n s,x l; nþ¼½ð1 x l ÞV a M ðyn s ; n 1Þ V sðy n s ; nþ X y b sš : ð36þ y n s The optimal exercise strategy y n s selected by the small firm target satisfies the first-order s ðy,y n s,x l; n s ¼ 0: ð37þ Solving this equation yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y n s ðx bx s r 2ðmþs l; nþ¼ 2 =2Þ b 2 ð1 x l ÞP a M ðn 1Þ P : ð38þ sðnþ This equation implies that, as a function of x l, the merger threshold y n s selected by the small firm target rises with x l. Fig. 2 charts y n s as a function of x l. In equilibrium, the negotiated ownership share must be such that the bidder and the target agree on merger timing, or y n l ¼ y n s. Using this condition, we can solve for the equilibrium timing and terms of the merger. The crossing point of the reaction functions in Fig. 2 characterizes the option exercise equilibrium. Proposition 6. Consider an asymmetric merger between a large firm and a small firm. Suppose that Assumptions 1 and 2 hold. (i) The value-maximizing merger policy is to merge when the industry shock ðy t Þ reaches the threshold value sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y n ðnþ¼ bðx s þx l Þ r 2ðmþs 2 =2Þ b 2 P a M ðn 1Þ P : lðnþ P s ðnþ ð39þ (ii) The merger threshold y n declines with a and n. (iii) The share of the merged firm accruing to the large firm is given by x n l ðnþ¼ X l X s þx l þ P lðnþx s P s ðnþx l ðx s þx l ÞP a M ðn 1Þ : ð40þ (iv) The ownership share x n l declines with n. Parts (i) and (ii) of Proposition 6 characterize the merger timing. As the values of the option to merge for both firms increase with the realization y of the industry shock, a merger occurs in a rising product market. Thus, consistent with empirical evidence documented by Maksimovic and Phillips (2001), cyclical product markets generate procyclical mergers. This result is also consistent with Mitchell and Mulherin s (1996) empirical finding that industry shocks contribute to the merger and restructuring activities during the 1980s.

11 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) As in most real options model, the merger threshold y n given in (39) determines the merger timing and merger likelihood. A higher value of the merger threshold implies a larger value of the expected time of the merger and a lower probability of merger within a given time horizon. 13 To interpret Eq. (39), we use Propositions 2 and 4 to rewrite it as V a M ðyn,n 1Þ V l ðy n ; nþ V s ðy n ; nþ¼ bðx s þx l Þ 4X s þx b 2 l : ð41þ Eq. (41) implies that, at the time of the merger, the benefit from the merger exceeds the sum of the merger costs X s þx l. This reflects the option value of waiting. Because mergers and acquisitions are analogous to an irreversible investment under uncertainty, the standard comparative statics results from the real options literature (e.g., Dixit and Pindyck, 1994) apply to merger timing. For example, an increase in the industry s demand uncertainty delays the timing of mergers, and an increase in the drift of the industry s demand shock speeds up the timing of mergers. Our novel comparative statics results are related to industry characteristics. First, part (ii) of Proposition 6 implies that the optimal merger threshold declines with the parameter a. Since the parameter a represents the exposure of the industry demand to the exogenous shock, one should expect to observe more mergers and acquisitions in industries where demand is more exposed to or more sensitive to exogenous shocks. The intuition behind this result is that an increase in the parameter a raises industry demand for a given positive shock. Thus, it increases anticompetitive gains P a M ðn 1Þ P l ðnþ P s ðnþ as shown in Lemma 1, thereby raising the benefits from merging. This result is consistent with empirical evidence in Mitchell and Mulherin (1996), who report that the industries experiencing the most merger and restructuring activity in the 1980s were the industries exposed most to industry shocks. We next turn to the effect of industry concentration on merger timing. Part (ii) of Proposition 6 implies that one should expect to see more mergers and acquisitions in more concentrated (i.e., less competitive) industries. The economic intuition behind this result is as follows. A higher level of pre-merger industry concentration is associated with higher anticompetitive profits by Lemma 1, which raises the incentive to merge, ceteris paribus, and hence the potential payoff from exercising the merger option. Importantly, higher anticompetitive profits at thetimeofamergerlead toahigheropportunity costofwaiting to merge, and so firms in less competitive industries will optimally exercise their option to merge earlier. This implication for the optimal timing of mergers in our Cournot Nash framework is in sharp contrast to most of the earlier findings in the literature on irreversible investment under uncertainty. Notably, Grenadier (2002) demonstrates that firms in more competitive industries will optimally exercise their investment options earlier in a symmetric industry equilibrium model. Intuitively, more industry competition increases the opportunity cost of waiting to invest and thus accelerates investment option exercise. We attribute the difference in our results to differences in the economic modeling of industry competition and structure. In our asymmetric industry equilibrium model, anticompetitive profits result from merging two firms to form a new firm, which alters product market competition endogenously. In Grenadier s (2002) model, anticompetitive profits result from exogenously reducing the number of identical firms that compete for an investment opportunity in the industry. Moreover, Grenadier (2002) studies an incremental investment problem, while we analyze a single discrete option exercise decision. We will consider scarcity of targets that can lead to competition among multiple bidders in Section 5. Bidder competition will hurt the acquirer, as is also shown by Morellec and Zhdanov (2005), and hence may attenuate the delayed option exercise due to the countervailing effect of product market competition that is central to our model. Parts (iii) and (iv) of Proposition 6 characterize the ownership share. Part (iii) shows that the large firm bidder demands a greater ownership share than the small firm target if the two firms incur identical merger costs X l ¼ X s. Intuitively, because the large firm has a higher pre-merger firm value, it demands a greater ownership share in our industry equilibrium. When the merger gains are endogenized the pre-merger industry concentration level also influences the ownership share. Part (iv) of Proposition 6 shows that a large merging firm demands a smaller ownership share in more concentrated industries, ceteris paribus. The intuition is that the pre-merger profit differential between the large and the small merging partners relative to the value of the merged firm declines with industry concentration. Thus, the large firm does not need to demand a greater share in more concentrated industries. In our model, total merger returns come from the merger surplus. The merger surplus is generated by two incentives to merge: (i) gaining market power, and (ii) reducing production costs. As Perry and Porter (1985) and Salant et al. (1983) note, the gain in market power alone may not be sufficient to motivate a merger. To illustrate this point, we decompose the total merger surplus (net of merger costs) evaluated at the equilibrium merger threshold y n into two components V a M ðyn ; n 1Þ V l ðy n ; nþ V s ðy n ; nþ X l X s ¼½ V ~ M ðy n ; nþ V l ðy n ; nþ V s ðy n ; nþ X l X s Š þ½v a fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} M ðyn ; nþ V ~ M ðy n ; nþš, ð42þ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} S 1 S 2 where ~ V M ðy n ; nþ represents the value of the merged firm when there is no cost saving. We compute this value using Proposition 1 by assuming that the merged firm uses the large firm s capital stock k to produce output and does not 13 The probability that a merger will take place in a time interval [0,T] is given by Pr sup YðtÞZy n ¼ N lnðy 0=y n Þþðm s 2 =2ÞT 0 rtrt s pffiffiffi þ y ð2m s 2 Þ=s 2 0 N lnðy 0=y n Þ ðm s 2 =2ÞT p T y n s ffiffiffi, T where N is the standard normal distribution function. This probability declines with the merger threshold y n.

12 596 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) Table 1 Decomposition of total merger surplus. The expressions V a M ðyn ; n 1Þ, V l ðy n ; nþ,andv sðy n ; nþ represent values of the merged firm, the pre-merger large firm, and the pre-merger small firm evaluated at the merger threshold. The expression V ~ M ðy n ; nþ represents the value of the merged firm evaluated at the merger threshold if it uses the large firm s capital stock to produce output. We decompose the merger surplus (net of merger costs) S V a M ðyn ; n 1Þ V sðy n ; nþ V l ðy n ; nþ X s X l into two components. One component, defined as S 1 ¼ V ~ M ðy n ; nþ V sðy n ; nþ V l ðy n ; nþ X s X l, is attributed to market power only. The other component, defined as S 2 ¼ V a M ðyn ; n 1Þ V ~ M ðy n ; nþ, is attributed to cost savings. Parameter values are K¼1, k¼0.2, b¼0.5, r¼0.08, m ¼ 0:01, s ¼ 0:20, and X l ¼ X s ¼ 2. Large firms V a M ðyn ; n 1Þ ~ V M ðy n ; nþ V sðy n ; nþ V l ðy n ; nþ S 1 S 2 S n¼2 183, ,376 61, ,053 53,076 53, n¼3 155, ,845 52, ,747 45,094 45, n¼4 135,527 96,333 45,345 90,147 39,163 39, combine the two merging firms capital assets to reduce production costs. Thus, the expressions S 1 and S 2 on the righthand side of Eq. (42) represent the merger surplus attributed to market power and cost savings, respectively. Table 1 illustrates these two components numerically for various values of n. One can see the effects of market power on the benefits (and hence returns) from mergers within our framework. First, note that the total merger surplus at the merger threshold is independent of n because it is equal to 2ðX s þx l Þ=b by Eq. (41). Second, the large firm and the small firm do not have incentives to merge if the motivation is market power alone. That is, the associated merger surplus (S 1 )is negative, even though the value of the merged firm is higher than the value of each merging firm. Third, it is the additional cost savings incentive represented by S 2 that makes the merger profitable. Finally, the table also reveals that the market power incentive is still present, although it does not motivate the merger by itself. This is because market power effect has a stronger effect as the industry becomes more concentrated (i.e., n is higher). Consequently, the component S 2 of the merger surplus generated by cost savings declines as the industry becomes more concentrated. Thus, the merger benefits in our numerical example are attributable to both cost savings and market power gains. Example 2. Fig. 3 illustrates Proposition 6. The input parameter values are a¼100, b¼0.5, K¼1, k¼0.2, n¼2, r¼8%, X l ¼ 2, X s ¼ 2, m ¼ 1%, and s ¼ 20%. Recall that Eq. (1) implies that the number of small firms satisfies m ¼ 2Kk 1 2n ¼ 6. Building on the insights from Example 1, the industry s parameter values a, b, K, k, and n, are selected such that a merger between two small firms is not profitable, i.e., P l ðnþ1þ 2P s ðnþo0, but a merger between a small and a large firm is profitable, i.e., P a M ðn 1Þ P sðnþ P l ðnþ40. Thus, Assumption 2 holds. To study the role of product market competition for mergers in isolation from frictions shared by other real options models of mergers, we use identical restructuring costs for the small firm target and the large firm acquirer. 14 That is, we set X f ¼ 2 for f¼s,l. The risk-free rate is taken from the yield curve on Treasury bonds. Similarly, the growth rate of industry shocks has been selected to generate a reasonable probability for a merger to arise in this industry. Finally, the value of the diffusion parameter has been chosen to match the time-series volatility of an average S&P 500 firm s asset return. As discussed earlier, the non-monotonicity of the profit differential with respect to the price sensitivity results from a tradeoff between a price effect and a quantity effect. Fig. 3a displays the impact of this economic phenomenon on the behavior of the merger threshold, n =@b. Moreover, because the merger gains increase with industry concentration when Assumption 2 is satisfied, the merger threshold decreases with the number of large firms in the industry, n, as shown in Fig. 3b. In particular, when n rises from 2 to 5, the merger threshold y n declines from 2.30 to If we set the initial industry shock to y 0 ¼ 1, these threshold values imply that the likelihood of a merger over a five-year horizon rises from 5.0% to 14.7%. Another interesting feature of the optimal merger threshold is that it is a decreasing function of k, as shown in Fig. 3c. Notice that the variance of firm size, k i 2fk=2,kg, in an industry with N firms is given by Varðk i Þ¼ ðk knþk 3 n 2ð2K knþ½2k kðnþ1þš, ð43þ which also increases with k. Thus, Fig. 3c and Eq. (43) reveal that, all else equal, takeovers are more likely in industries in which firm size is more dispersed. Although we are unable to prove this result theoretically, we have verified that it holds true for a wide range of parameter values. Intuitively, the endogenous synergy gains from merging increase when the size difference of the two merging partners tangible assets rises. Finally, and as supported by Fig. 3, other standard comparative statics results apply within our model, so we do not discuss them Cumulative returns We now turn to cumulative returns resulting from an asymmetric merger. The equity value of a large merging firm before a merger, denoted E l ðyðtþ; nþ, is equal to the value of its assets in place plus the value of the merger option E l ðyðtþ; nþ¼v l ðyðtþ; nþþom l ðyðtþ,y n,x n l ; nþ, ð44þ 14 Intuitively, larger restructuring costs delay the timing of mergers. We will relax this assumption in Section 5.

13 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) Fig. 3. Merger timing. This figure plots the merger threshold y n as a function of the price sensitivity of demand, b, the number of large firms, n, the size of the large firm s tangible asset, k, the risk-free rate, r, the growth rate of industry shocks, m, and the volatility of industry shocks, s. Parameter values are a¼100, b¼0.5, K¼1, k¼0.2, n¼2, r¼8%, X l ¼ 2, X s ¼ 2, m ¼ 1%, and s ¼ 20%. where OM l ðyðtþ,y n,x n l ; nþ is given in Eq. (32), and yn and x n l are given in Proposition 6. Similarly, the equity value of a small merging firm before the merger, denoted E s ðyðtþ; nþ, is given by E s ðyðtþ; nþ¼v s ðyðtþ; nþþom s ðyðtþ,y n,x n l ; nþ, ð45þ where OM s ð; nþ is defined in Eq. (36). We may express the cumulative stock returns as a fraction of the stand-alone equity value V f ðyðtþ; nþ of the small firm f¼s and the large firm f¼l. That is, the cumulative returns to the small and large merging firms at time t rt y n are given by R f,m ðyðtþ,nþ¼ E f ðyðtþ; nþ V f ðyðtþ; nþ V f ðyðtþ; nþ ¼ OM f ðyðtþ,y n,x n l ; nþ, ð46þ V f ðyðtþ; nþ for f ¼ s,l. The cumulative return to the merging firm f at the time of the merger announcement is equal to the expression in Eq. (46) evaluated at t ¼ t y n or YðtÞ¼y n.

14 598 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) Similarly, we can compute the cumulative stock return to a rival firm at the time of the merger announcement. To do so, we first compute the equity value of a rival firm prior to the announcement of a merger at date trt y n. It is equal to the value of assets in place before the merger plus an option value from the merger V f ðyðtþ; nþþe y ½e rðt y n tþ ðv a f ðyðt y nþ; n 1Þ V f ðyðt y nþ; nþþ9yðtþš, ð47þ for f ¼ s,l. This option value results from the fact that the value of the rival firm becomes V a ðyðt f ynþ; n 1Þ after the asymmetric merger since there are n 1 large firms, m 1 small firms, and a huge merged firm in the industry. We then define the cumulative return to a small or large rival firm before the merger as R f,r ðyðtþ; nþ¼ Ey ½e rðtyn tþ ðv a ðyðt f y nþ; n 1Þ V f ðyðt y nþ; nþþ9yðtþš, ð48þ V f ðyðtþ; nþ for f ¼ s,l. We focus on the cumulative return at the time of the merger announcement, when t ¼ t y n or Yðt y nþ¼y n. Proposition 7. Consider an asymmetric merger between a large firm and a small firm. Suppose that Assumptions 1 and 2 hold. (i) The cumulative stock returns to the small and large merging firms at the time of restructuring are given by R f,m ðy n ; nþ¼ Pa M ðn 1Þ P lðnþ P s ðnþ X f 2 P f ðnþ X s þx l b, ð49þ for f ¼ s,l. (ii) The cumulative stock returns to a small or a large rival firm at the time of restructuring are given by " # R f,r ðy n ; nþ¼ 1 b þ DðnÞ 1, ð50þ for f ¼ s,l. (iii) All the above returns are positive and increase with n. Proposition 7 highlights several interesting aspects of cumulative stock returns within our Cournot Nash industry equilibrium framework. First, Eq. (49) reveals that the cumulative returns to the two merging firms have three types of determinants, including an anticompetitive effect, two size effects, and a hysteresis effect. The hysteresis effect is represented by 2=b, which is a function of the risk-free rate, r, the growth rate of industry shocks, m, and the volatility of industry shocks, s. It implies that a higher volatility of industry shocks, s, leads to higher cumulative returns to both the acquirer and the target. With more uncertainty surrounding the industry, the merger option is exercised when it is deeper in the money, resulting in higher cumulative stock returns. The two size effects are captured by 1=P f ðnþ and X f =ðx s þx l Þ, which respectively reflect the results that the smaller merging firm or the firm with a higher merger cost earns a higher return than the larger merging firm or the firm with a lower merger cost. Second, and unlike existing real options models of mergers, our model with constant returns to scale does not produce an exogenous synergy effect for merger returns. Instead, we have an anticompetitive and cost reduction effect represented by the term P a M ðn 1Þ P lðnþ P s ðnþ, which characterizes the merger benefits as a function of industry equilibria before and after the control transaction. This effect reflects the fact that after a merger, there are fewer small and large firms in the industry, and a huge merged entity emerges. Hence, both the market structure and the competitive landscape of the industry change at the time of the merger. In addition, the huge merged firm combines the tangible assets of the two merging firms, thereby reducing production costs. The fraction ½P a M ðn 1Þ P lðnþ P s ðnþš=p f ðnþ in Eq. (49) commingles the first size effect with the anticompetitive effect, indicating the percentage gain in profits resulting from a control transaction. Notably, this percentage gain differs for a small and a large merging firm as it depends on the firm s size and hence its relative contribution to the merger benefits. Third, merging firms earn higher returns in more concentrated industries. The intuition is that there is a stronger anticompetitive effect on the equilibrium price after a merger in those industries. As a consequence, merging firms derive higher returns from entering into a takeover. Fourth, returns to merger and rival firms are invariant to the sensitivity to industry-wide demand shocks. Recall that only the anticompetitive effect of returns depends on a. By inspection of Eqs. (25) (27), notice that all firms profit multipliers are a quadratic function of a. Thus, even though exposure to industry-wide shocks crucially affects the optimal timing of mergers, it does not affect the returns from restructuring decisions in our industry equilibrium framework. Fifth, Eq. (50) shows that the cumulative return to a small or a large rival firm at the time of a merger announcement is positive too. Like the cumulative returns to merging firms, the cumulative returns to rival firms increase with industry concentration. The intuition is that the industry s equilibrium price rises after the merger, and rival firms also benefit from this price increase. This benefit increases with industry concentration. Like merger returns, rival returns are therefore higher in more concentrated industries. Notice, however, that the magnitude of rival returns does not depend on firm size, as rival firms do not change their capital stock at the time of a control transaction but only adjust their equilibrium output choices. Example 3. We illustrate Proposition 7 by a numerical example with baseline parameter values used in Examples 1 and 2. Recall that we let X l ¼ X s to focus on the portion of merger returns that is due not simply to different restructuring costs, which is also a feature of other real options models of mergers and hence not unique to our framework. Fig. 4 graphs the

15 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) Fig. 4. Cumulative stock returns. This figure depicts the cumulative stock returns of a small merging firm (dotted line), a large merging firm (dashed line), and a rival firm (solid line) as a function of the price sensitivity of demand, b; the number of large firms, n; the size of the large firm s tangible asset, k; the risk-free rate, r; the growth rate of industry shocks, m; and the volatility of industry shocks, s. Parameter values are a¼100, b¼0.5, K¼1, k¼0.2, n¼2, r¼8%, X l ¼ 2, X s ¼ 2, m ¼ 1%, and s ¼ 20%. cumulative returns of a small merging firm (dashed line) and a large merging firm (dotted line) given in Eq. (49) as well as the return to a rival firm (solid line) given in Eq. (50) as functions of various industry characteristics, such as the price sensitivity of demand, b, and the number of large firms, n. The figure reveals several interesting aspects of the determinants of merger and rival returns, such as firm size, profitability, and the firm s (relative) contribution to the creation of the merger benefit. First, and somewhat surprisingly, the return of a small merging firm (i.e. target) exceeds the return of a large merging firm (i.e. acquirer). Mathematically, the anticompetitive effect, P a M ðn 1Þ P lðnþ P s ðnþ, is scaled by the firm s pre-merger profitability or status quo, P f ðnþ, in Eq. (49), so that merger returns reflect a percentage gain in profits that differs for small and large merging firms. Crucially, this percentage gain depends on the pre-merger firm size and hence the firm s relative contribution to the post-merger synergy gains. The intuition for this interesting finding is therefore that the smaller firm benefits relatively more from a merger in our industry equilibrium framework and hence enjoys a larger relative synergy gain (i.e. merger return). As mentioned earlier, mergers typically involve acquisitions of a small firm by a large firm. This implication of the model therefore supports the available evidence without reliance on additional, behavioral assumptions, such as misvaluation (see, e.g., Shleifer and Vishny, 2003).

16 600 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) Second, merger returns vary non-monotonically with the price sensitivity of demand, b, which is consistent with our findings in the earlier examples. Yet, rival returns increase with the price sensitivity of demand, b. The reason for this interesting difference between merger and rival returns is that industry rivals benefit from the (positive) price effect but are not hurt by the (negative) quantity effect of the Cournot Nash equilibrium. In fact, because rivals do not change their firm size, k i, they optimally produce a slightly higher quantity subsequent to a successful takeover deal. Third, the graph in Fig. 4b studies the effect of the number of large firms, n, on merger and rival returns. The figure reveals that a higher level of concentration in the industry (i.e. a larger value of n) leads to higher returns for all firms in the industry. Fourth, merger and rival returns are increasing in k, as shown in Fig. 4c. Recall that the variance of firm size in Eq. (43) also increases with k. Thus, returns to merging and rival firms arising from restructuring are higher in industries with more dispersion in firm size. Finally, and as confirmed by Fig. 4, other comparative statics results are largely due to the hysteresis effect that is shared by other real options models of mergers, so we do not discuss them. 5. Merger equilibrium with multiple bidders So far, we have focused on an asymmetric merger with a single bidder. We now consider two bidders competing for a small firm target and suppose Assumption 3 holds throughout this section, so that both bidders have incentives to merge with the target. We consider two cases. In the first case, the two bidders are different in that one is a large firm and the other is a small firm. In the second case, the two bidders are identical in that either both are large firms or both are small firms. Intuitively, bidder competition puts the target into an advantageous position and allows target shareholders to extract a higher premium from the bidding firms. Notably, we do not rely on, e.g., asymmetric information for bid premium to obtain. Merger benefits are derived endogenously from an industry equilibrium (under perfect information) rather than exogenously specified. Thus, product market competition may interact with bidder competition Two different bidders Suppose that one bidder is a large firm and the other is a small firm. The bidding game is as follows. Once the contest is initiated, the two bidding firms submit bids to the target in the form of a fraction of the merged firm s equity to be owned by target shareholders upon the takeover. The bidder who offers, in present value terms, the highest monetary payoff to the target shareholders wins the contest. Given the winner s ownership share, the winning bidder and the target select their merger timing independently. In equilibrium, they must agree on the merger timing. 15 In our model, a large firm has a production cost advantage over a small firm. Thus, the value of the merged firm is higher when the small firm target merges with a large firm bidder than when it merges with a small firm bidder. Formally, we can use Eqs. (17) and (27) to show that V a M ðy; n 1Þ4V lðy; nþ1þ: ð51þ Eq. (51) implies that the large firm will win the takeover contest as long as the takeover is profitable to it because it can always slightly bid more than the small firm bidder and deliver more value to target shareholders. This strategy is costly to the large firm when its merger costs are sufficiently higher than the small firm bidder s merger costs. In this case, the takeover may not be profitable to the large firm bidder, and it would rather drop out. From the preceding discussions, we conclude that either the large firm or the small firm may win the takeover contest, depending on the relative effects of production and merger costs. In the analysis below, we will focus only on the empirically more plausible case where the large firm wins the contest. The analysis of the case where the small firm wins the contest is contained in a previous version of this paper and is available upon request. Before presenting our analysis, we define the breakeven share x BE s for the small firm bidder as x BE s V lðy; nþ1þ V s ðy; nþ X s ¼ 0: ð52þ If the small firm bidder places a bid higher than 1 x BE s, then it will realize a negative value by entering the deal. In this case, the small firm bidder would be better off losing the takeover contest. Similarly, we define the breakeven share x BE l for the large firm bidder as x BE l V a M ðy; n 1Þ V lðy; nþ X l ¼ 0: ð53þ We now discuss the bidders strategies when the large firm bidder wins the contest. In this case, the losing small firm bidder may still influence equilibrium, depending on whether it is strong or weak. As a result, there are two possibilities to consider: 1. The losing small firm bidder is weak in the sense that merging with the large firm bidder and accepting the ownership share 1 x n l is more profitable to the target than merging with the small firm bidder and accepting the ownership 15 Our analysis follows similar steps as in Morellec and Zhdanov (2005), in which the target cares about the monetary value of a bid. We also use ownership share as a device to solve the model. Any monetary value of a bid can be transformed into a corresponding ownership share by dividing the monetary value of the bid by the combined firm value.

17 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) share 1 x BE s : ð1 x n l ÞV a Mðy; n 1Þ4ð1 xbe s ÞV lðy; nþ1þ, ð54þ where x n l is the equilibrium share without bidder competition given in Proposition 6. For the small firm bidder to win, it must offer an ownership share of at least 1 x BE s to target shareholders, which implies a negative net payoff to the small firm bidder. Thus, it would rather drop out of the takeover contest and the equilibrium is the same as in Proposition The small firm bidder is strong in the sense that merging with the large firm bidder and accepting the ownership share 1 x n l is less profitable to the target than merging with the small firm bidder and accepting the ownership share 1 x BE s : ð1 x n l ÞV a Mðy; n 1Þoð1 xbe s ÞV lðy; nþ1þ: ð55þ Under this condition, the small firm bidder has an incentive to bid an amount slightly less than 1 x BE s in order to win the contest. Anticipating this bidder s competition, the large firm bidder will place a bid higher than 1 x n l until equality holds in Eq. (55). We define x max l as the ownership share satisfying this equality: ð1 x max l ÞV a Mðy; n 1Þ¼ð1 xbe s ÞV lðy; nþ1þ: ð56þ The value x max l is the maximum share that the large firm bidder can extract from the merged firm such that it still wins the takeover contest. If the large firm demands a share higher than x max l, or places a bid lower than 1 x max l, the small firm will outbid the large firm and win the contest. Thus, the best response of the large firm is to place a bid 1 x max l. Note that the ownership share x max l of the large firm bidder must be greater than the breakeven share x BE l for the large firm to participate. If this condition is violated, the large firm would rather drop out. Given the above bidding strategies, the winning bidder will select a merger time to maximize the option value to merge. The target shareholders also select a merger time to maximize their option value to merge, given the ownership share proposed by the winning bidder. The equilibrium ownership share is determined so that the merger timing selected by the winning bidder and the target shareholders agrees. Fig. 5 illustrates equilibria when the large firm bidder wins the takeover contest. The increasing function represents the target s strategy. The solid (dashed) decreasing line represents the large firm bidder s strategy when the small firm bidder is weak (strong). In particular, the dashed line represents y as a function of x max l derived from Eq. (56) after we substitute x BE s from Eq. (52). The crossing points of the reaction functions in Fig. 5 represent option exercise equilibria, which are formally presented in Proposition 8. Proposition 8. Consider a small firm bidder and a large firm bidder competing for a small firm target. Suppose Assumptions 1 and 3 hold. Let LðnÞ Pa M ðn 1Þ P lðnþ P s ðnþ P l ðnþ1þ 2P s ðnþ ð57þ denote the relative merger benefits. (i) If X l o 2ðb 1ÞLðnÞ 1, ð58þ X s b then the equilibrium is the same as that with a single large firm bidder described in Proposition 6. The cumulative stock returns to the winning large firm bidder and the small firm target are given in Propositions 7. (ii) If 2ðb 1ÞLðnÞ 1o X l o 2ðb 1ÞLðnÞ b b X s b 2 b 2, ð59þ Fig. 5. Reaction functions with multiple bidders. The figure plots the reaction functions of the acquirer and the target, as a function of the bidder s ownership share. The increasing line represents the target s strategy. The solid (dashed) decreasing line represents the large firm bidder s strategy when the losing bidder is weak (strong). The crossing points of the reaction functions represent the option exercise equilibria of Proposition 8.

18 602 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) then the large firm bidder wins the contest and the small firm bidder is strong. The takeover takes place the first time the industry shock ðy t Þ reaches the threshold value sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y n cl ðnþ¼ 2ðb 1Þ X s ðr 2ðmþs 2 =2ÞÞ : ð60þ ðb 2Þ P l ðnþ1þ 2P s ðnþ The share of the merged firm accruing to the winning large firm bidder is given by x max l ðnþ¼1 bp lðnþ1þ 2P s ðnþ 2ðb 1ÞP a M ðn 1Þ : The cumulative returns to the winning large firm bidder and the small firm target are given by R l,m ðy n cl ; nþ¼ Pa M ðn 1Þ P lðnþ P s ðnþ P l ðnþ ðb 2ÞX l þbx s P l ðnþ1þ 2P s ðnþ 2ðb 1ÞX s P l ðnþ ð61þ ð62þ and R s,m ðy n cl ; nþ¼ P lðnþ1þ 2P s ðnþ : ð63þ ðb 1ÞP s ðnþ Proposition 8 provides a characterization of two possible merger equilibria with different bidders. While, in principle, the small firm bidder could win the takeover contest if it had a large enough merger cost advantage, we only consider equilibria in which the large firm bidder always wins the takeover contest. We believe that this case is more relevant in corporate practice because merger costs include fees to investment banks and lawyers, and should not vary substantially for firms of different sizes (see Gorton et al., 2009). 16 The two conditions (58) and (59) measure the relative cost advantage and the relative merger benefit. They are the key to determining the equilibrium outcomes in the presence of bidder competition. Part (i) of Proposition 8 shows that if the small firm bidder s merger cost advantage is not large enough in that condition (58) holds, then its competition with the large firm bidder does not matter for the large firm bidder s behavior. As a result, the equilibrium is the same as that with a single large firm bidder described in Proposition 6. Part (ii) of Proposition 8 establishes that if the small firm bidder s merger cost advantage is sufficiently large, then its competition with the large firm bidder allows the target to extract additional returns from the winning bidder. That is, if condition (59) holds, the merger return to the winning large firm bidder given in Eq. (62) is lower than without the bidder competition given in Eq. (49), while the merger return to the small target firm in Eq. (63) is greater than without bidder competition given in Eq. (49). To interpret Proposition 8 further, we define the bid premium resulting from bidder competition as the percentage increase in the target equity value over the case without bidder competition. These equity values are evaluated at the equilibrium time of the merger. Clearly, there is a positive bid premium only if the losing bidder is strong as in part (ii). In this case, we can formally define the bid premium resulting from bidder competition as ½ð1 x max l ÞV a M ðyn cl ; n 1Þ ð1 xn l ÞVa M ðyn cl ; n 1ÞŠ ð1 x n l ÞV a M ðyn cl ; n 1Þ ¼ xn l xmax l : ð64þ 1 x n l We can use condition (59) and the expressions for x n l and x max l to demonstrate that x n l 4xmax l. Consequently, the bid premium is positive. Note that both condition (59) and the bid premium depend on industry characteristics, such as the industry concentration level, n, and the price sensitivity of demand parameter, b. As a result, there is an interaction between product market competition and bidder competition in our model. Example 4. A numerical example illustrates the results reported in Proposition 8, focusing on the interesting interactions between bidder competition and industry competition. Baseline parameter values are: a¼100, k¼0.2, K¼1, b¼0.4, n¼4, r¼8%, m ¼ 1%, and s ¼ 20%. To ensure that the large firm bidder wins the takeover contest and the small firm bidder can be potentially strong as in part (ii) of Proposition 8, we set X l ¼ 20 and X s ¼ 1 so that condition (59) holds. Fig. 6 displays the bid premium in Eq. (64) as a function of the number of large firms, n, and the price sensitivity of demand, b. The figure highlights several key elements of the different equilibria resulting from jointly analyzing bidder competition and product market competition. First, the left panel shows that the bid premium is first an increasing and then a declining function of the price sensitivity parameter b. When b is low enough, the bid premium is close to zero. When b is high enough, condition (58) holds. As a result, the small firm bidder does not influence the equilibrium bidding process, and there is no bid premium We have also noted earlier that large firm bidders and small firm targets are consistent with the empirical evidence. For example, Moeller et al. (2004) report that contested deals are more likely for large acquirers than for small acquirers, which implies that large firm bidders tend to win more often in takeover contests. 17 While the bid premium resulting from the interaction of bidder and industry competition in our setting can be higher or lower for different parameter values, it is only one component of the combined run-up and markup observed in corporate practice.

19 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) Fig. 6. Bidder competition and industry competition. The figure presents the bid premium ðx n l xmax l Þ=ð1 x n l Þ as a function of the number of large firms n and the price sensitivity of demand b. Parameter values are a¼100, b¼0.4, K¼1, k¼0.2, n¼4, r¼8%, X l ¼ 20, X s ¼ 1, m ¼ 1%, and s ¼ 20%. Second, the right panel of the figure illustrates our earlier discussion that only when the industry is sufficiently concentrated, bidder competition matters. In the numerical example, bidder competition induces an additional bid premium if there are at least three large firms in the industry. Moreover, in the region where the bid premium is non-trivial it increases with industry concentration. Absent of a formal model, one might have been tempted to conclude that bidder competition generally raises the price of a contested deal irrespective of industry competition. However, this example underlines that the degree of industry competition may play a central role for if and how a bid premium emerges in equilibrium Two identical bidders We now consider the alternative case in which there are two identical firms competing for a small target. We suppose that both bidders are large firms. Hence the production cost advantage from the previous subsection with different bidders is not present in this subsection. The analysis of the case with two identical small firms is similar and hence is omitted here. Without loss of generality, we label the two large firms as firm 1 and firm 2. From the preceding subsection, we conclude that either firm 1 or firm 2 may win the takeover contest. We conjecture that the equilibrium bid must be equal to x BE l in Eq. (53), so that each bidder wins with equal probability and earns zero surplus. The intuition is as follows. Suppose firm 1 wins the contest and obtains ownership share x 1. To earn nonnegative surplus, it must be the case that x 1 Zx BE l.butifx 1 4x BE l,thenfirm2can offer ownership share 1 x BE l e to the target for a sufficiently small e40. The target s payoff if merging with firm 2 will be ð1 x BE l eþv a Mðy; n 1Þ, ð65þ which is greater than the payoff if merging with firm 1 for a sufficiently small e40 ð1 x 1 ÞV a Mðy; n 1Þ: ð66þ Thus, firm 2 will win the contest and still earns a positive surplus. This contradicts with our previous supposition. We deduce that, in equilibrium, the winning large firm will bid its breakeven share x BE l. The next proposition provides the solution of the takeover contest with two identical, large bidders. Proposition 9. Consider two large firm bidders competing for a small firm target. Suppose Assumptions 1 and 2 hold. (i) The takeover takes place the first time the industry shock ðy t Þ reaches the threshold value sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bx s ½bX s þðb 2ÞX y % ðnþ¼ l Š½r 2ðmþs 2 =2ÞŠ b 2 bx s ½P a M ðn 1Þ P : ð67þ lðnþ P s ðnþš (ii) The share of the merged firm accruing to the winning large firm bidder is given by x % ðnþ¼ X lðb 2Þ½P a M ðn 1Þ P sðnþšþbx s P l ðnþ ½bX s þðb 2ÞX l ŠP a M ðn 1Þ : ð68þ (iii) The cumulative returns to the winning large firm bidder and the small firm target are given by R l,m ðy % ; nþ¼0 ð69þ and R s,m ðy % ; nþ¼ 2X s½p a M ðn 1Þ P lðnþ P s ðnþš : ð70þ ½bX s þðb 2ÞX l ŠP s ðxþ Proposition 9 establishes the results for merger timing, merger terms, and merger returns for the case of competition among two identical (large) bidders. In this case, competition is fiercer than in the case of different bidders in the previous

20 604 D. Hackbarth, J. Miao / Journal of Economic Dynamics & Control 36 (2012) section, which puts the target into an even more advantageous position and hence produces even higher returns to target shareholders. Importantly, the two identical bidders competition squeezes away all surplus to the two bidders and allows the target to extract all merger surplus. 6. Conclusion This paper develops a real options model of mergers and acquisitions that jointly determines the industry s product market equilibrium and the timing and terms of takeovers. The analysis explicitly recognizes the role of product market competition and derives equilibrium restructuring strategies by solving an option exercise game between bidding and target shareholders. Our results are generally consistent with the extant empirical evidence. In addition, our model also generates a number of novel empirical implications: 1. Product market competition among heterogeneous firms does not speed up the acquisition process. Instead, we find that increased industry competition delays the timing of takeovers. Thus, mergers are more likely in more concentrated industries. 2. The relative scarcity of acquirers and targets in the industry is reflected in the equilibrium equity share (i.e., merger terms). In particular, a large acquirer (small target) demands a lower (higher) ownership share in the merged firm if the industry is less competitive. 3. Merger returns are higher for the smaller merging firm (e.g. target) than the larger merging firm (e.g. acquirer) when the firms have identical merger costs. Returns to merging and rival firms arising from restructuring are higher in more concentrated industries. 4. Takeovers are more likely in industries in which dispersion in firm size is higher. In addition, returns to merging and rival firms arising from restructuring are higher in industries with more dispersion in firm size. 5. With multiple bidders for a scarce target and similar merger costs, the large firm bidder wins the takeover contest when competing with a small firm bidder. Only in a sufficiently concentrated industry, bidder competition speeds up the takeover process and leads the large firm acquirer to pay a bid premium to discourage the competing small firm competitor. If it exists, this bid premium decreases with industry competition. By contrast, in a sufficiently competitive industry, the small firm bidder does not matter, and the equilibrium outcome corresponds to the one without bidder competition. 6. If the two bidding firms are identical, then bidder competition will squeeze away all surplus to the bidders and allow target shareholders to extract all merger surplus. The rich set of predictions generated by integrating a real options model of mergers into an industry equilibrium framework suggests that further extensions of our model will be valuable. Some possible extensions would add complexity to our analysis by refining the link between industry structure and takeover activity. For example, our analysis follows most research in the literature by assuming exogenously the initiation of mergers in the sense that the merger structure (i.e., what firm merges with what firm, and which remains independent) is exogenously imposed in the absence of antitrust law. As a result, our model indicates that industry rivals benefit from mergers without having to bear the restructuring costs, leading to a so-called free-rider problem, i.e., merging firms provide a positive externality to rival firms. This free rider problem may give firms incentives to further delay takeovers in more competitive industries. More generally, there may be a race to merge in that multiple firms compete to merge with one firm or multiple other firms. In this case, merging firms may face a preemption threat, which would be technically challenging within our setup. As is well known in the literature on the continuous-time models of preemption games, one has to define strategies appropriately and address the issue of coordination failures (see, e.g., Simon and Stinchcombe, 1989). Some researchers study real options models of preemption games extending an early contribution by Fudenberg and Tirole (1985). Weeds (2002) presents a model in which two firms invest in competing research projects with uncertain returns. She assumes that successful innovation by one firm eliminates all possible profits for the other, giving rise to a first-mover advantage. Mason and Weeds (2010) study an irreversible investment problem in which two firms compete for a project. There is an advantage to being the first to invest and externalities to investing when the other firm also does so. For preemption games with two firms, there are typically two types of equilibria. In the first type, the two firms invest simultaneously. In the second type, the two firms invest sequentially. Argenziano and Schmidt-Dengler (2007) extend Fudenberg and Tirole s (1985) duopoly model to an arbitrary, finite number of firms. Their analysis is quite complex. In their case, each firm s investment time may be different, but it is also possible that some firms investment times are clustered. One could adapt the analysis of Argenziano and Schmidt-Dengler (2007) to our setup with many firms. However, the solution would be more complex because we would have to consider each pair s merger decisions taking into account the endogenous change of industry structure after a merger. Furthermore, we have to consider mergers when each firm makes individual merger decisions and responds to mergers by other firms, which is a difficult problem even in a static model (see, e.g., Qiu and Zhou, 2007). In this case, multiple mergers may arise sequentially, and hence the order of mergers may be an important element of the dynamic evolution of the industry s competitive landscape through time. Similarly, modeling new entry into and/or exit from the industry